Decomposability and triangularizability of positive operators on Banach lattices.
Date
1997
Authors
Jahandideh, Mohammad Taghi.
Journal Title
Journal ISSN
Volume Title
Publisher
Dalhousie University
Abstract
Description
The main results in this thesis are about invariant subspaces of multiplicative semigroups of quasinilpotent positive operators on Banach lattices.
There are some known results that guarantee the existence of a non-trivial closed invariant ideal for a quasinilpotent positive operator on certain spaces, for example on $C\sb0(\Omega)$ with $\Omega$ a locally compact Hausdorff space or on a Banach lattice with atoms. Some recent results also guarantee the existence of non-trivial closed invariant ideals for a compact quasinilpotent positive operator on an arbitrary Banach lattice. In fact it is known that given such an operator T, on a real or complex Banach lattice, there is a nontrivial closed ideal which is invariant under all positive operators that commute with T.
This thesis deals with invariant ideals for families of positive operators on Banach lattices. In particular it studies ideal-decomposable and ideal-triangularizable semi-groups of positive operators. We show that in certain Banach lattices compactness is not required for the existence of hyperinvariant closed ideals for a quasinilpotent positive operator. We also show that in those Banach lattices a semigroup of quasinilpotent positive operators might be decomposable without imposing any compactness condition. We generalize the fact that the only irreducible $C\sb{p}$-closed subalgebra of $C\sb{p}$ is $C\sb{p}$ itself to extend some recent reducibility results and apply them to derive some decomposability theorems concerning a collection of quasinilpotent positive operators on reflexive Banach lattices.
We use these results for "ideal-triangularization", i.e., we construct a maximal closed ideal chain, each of whose members is invariant under a certain collection of operators that are related to compact positive operators, or to quasinilpotent positive operators.
Thesis (Ph.D.)--Dalhousie University (Canada), 1997.
There are some known results that guarantee the existence of a non-trivial closed invariant ideal for a quasinilpotent positive operator on certain spaces, for example on $C\sb0(\Omega)$ with $\Omega$ a locally compact Hausdorff space or on a Banach lattice with atoms. Some recent results also guarantee the existence of non-trivial closed invariant ideals for a compact quasinilpotent positive operator on an arbitrary Banach lattice. In fact it is known that given such an operator T, on a real or complex Banach lattice, there is a nontrivial closed ideal which is invariant under all positive operators that commute with T.
This thesis deals with invariant ideals for families of positive operators on Banach lattices. In particular it studies ideal-decomposable and ideal-triangularizable semi-groups of positive operators. We show that in certain Banach lattices compactness is not required for the existence of hyperinvariant closed ideals for a quasinilpotent positive operator. We also show that in those Banach lattices a semigroup of quasinilpotent positive operators might be decomposable without imposing any compactness condition. We generalize the fact that the only irreducible $C\sb{p}$-closed subalgebra of $C\sb{p}$ is $C\sb{p}$ itself to extend some recent reducibility results and apply them to derive some decomposability theorems concerning a collection of quasinilpotent positive operators on reflexive Banach lattices.
We use these results for "ideal-triangularization", i.e., we construct a maximal closed ideal chain, each of whose members is invariant under a certain collection of operators that are related to compact positive operators, or to quasinilpotent positive operators.
Thesis (Ph.D.)--Dalhousie University (Canada), 1997.
Keywords
Mathematics.